Skip to main content

On the Existence of Rainbow 4-Term Arithmetic Progressions

Abstract

For infinitely many natural numbers n, we construct 4-colorings of [n]  =  {1, 2, ..., n}, with equinumerous color classes, that contain no 4-term arithmetic progression whose elements are colored in distinct colors. This result solves an open problem of Jungić et al. (Comb Probab Comput 12:599–620, 2003) Axenovich and Fon-der-Flaass (Electron J Comb 11:R1, 2004).

This is a preview of subscription content, access via your institution.

References

  • Axenovich, M., Fon-Der-Flaass, D.: On rainbow arithmetic progressions. Electron. J. Comb. 11, R1 (2004)

  • Landman, B., Robertson, A.: Ramsey theory on the integers. American Mathematical Society. Student Mathematical Library 24 (2003)

  • Conlon, D.: Rainbow solutions of linear equations over \({\mathbb{Z}}_{p}\) . Discrete Math. 306, 2056–2063 (2006)

    Google Scholar 

  • Graham, R. L., Rothschild, B. L., Spencer, J. H.: Ramsey theory. Wiley, New York (1990)

  • Jungić, V., Licht (Fox), J., Mahdian, M., Nešetřil, J., Radoičić, R.: Rainbow arithmetic progressions and anti-Ramsey results. Comb. Probab. Comput. 12, 599–620 (2003)

    Google Scholar 

  • Jungić, V., Radoičić, R.: Rainbow 3-term arithmetic progressions. Integers Electron. J. Combin. Number Theory 3, A18 (2003)

  • Szemerédi, E.: On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27, 199–245 (1975)

    Google Scholar 

  • van der Waerden, B. L.: Beweis einer Baudetschen Vermutung. Nieuw Archief voor Wis. 15, 212–216 (1927)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Radoš Radoičić.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Conlon, D., Jungić, V. & Radoičić, R. On the Existence of Rainbow 4-Term Arithmetic Progressions. Graphs and Combinatorics 23, 249–254 (2007). https://doi.org/10.1007/s00373-007-0723-2

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-007-0723-2

Keywords

  • Arithmetic Progression
  • Distinct Color
  • Color Class
  • Acta Arith
  • Consecutive Integer